Employers and clients want to know that we will be able to work energetically for them. And get results. Likely, they don't care much about our personality. That usually only becomes a factor when we present ourselves as difficult to work with.
This isn't new. Arthur Miller depicted that meme in his iconic drama "Death of a Salesman." Willy Loman lost his focus on the prospect. Instead he turned inward and locked himself into assuming the cult of personality could and would close the sale.
Sure, there have always been those who get breaks through their charisma. Bill Clinton heads the list. But his wife, Hillary Clinton, who is not likable, may wind up accomplishing more for America than he had.
The bottom line on this is: In our cover letters we want to send the message that we are the kind who will pull out all stops in getting results and have the track record that proves that.
If we move on to the next stage - the interview - again, we want to transmit those two memes. We will make sure our personality isn't an obstacle. Otherwise, there should be no focus on the self.